-Portfolio are combinations of underlying assets
-Given return properties of the underlying assets, what are the return properties of their combination?
Expected Return of Portfolio of \(N\) Assets
\[ E[r_p] = \sum_{i=1}^{N} w_i E[r_i] \]
-\(w_i\) is the portfolio weight of asset \(i\)
-\(E[r_i]\) is the expected return of asset \(i\)
-The portfolio is fully invested: \(\sum_i w_i = 1\)
-Notation: \(E(r_p)=\mu_i\)
-\(w_i\) is the portfolio weight of asset \(i\)
-\(\text{cov}[r_i,r_j]\) is the covariance between assets \(i\) and \(j\)
-Recall that \(\text{cov}[r_i,r_j]=\text{var}[r_i]\) and \(\text{sd}[r_i]=\sqrt{\text{var}[r_i]}\)
-Notation: \(\text{var}[r_p]=\sigma^2_p\); \(\text{cov}[r_i,r_j]=\sigma_{i,j}\); \(\text{sd}[r_p]=\sigma_p\)
Variance of Portfolio of \(N\) Assets: A Matrix View
Claim: The variance of the return of a portfolio with many securities depends more on the covariances between the individual securities than on the variances of the individual securities.
\(w_1^2 \text{var}[r_1]\)
\(w_1 w_2 \text{cov}[r_1,r_2]\)
\(w_1 w_3 \text{cov}[r_1,r_3]\)
\(w_2 w_1 \text{cov}[r_2,r_1]\)
\(w_2^2 \text{var}[r_2]\)
\(w_2 w_3 \text{cov}[r_2,r_3]\)
\(w_3 w_1 \text{cov}[r_3,r_1]\)
\(w_3 w_2 \text{cov}[r_3,r_2]\)
\(w_3^2 \text{var}[r_3]\)
Now consider a five-security portfolio
How many variance terms?
\[N=5\]
How many covariance terms?
\[N^2-N = 25-5=20 \]
\(w_1^2 \text{var}[r_1]\)
\(w_1 w_2 \text{cov}[r_1,r_2]\)
\(w_1 w_3 \text{cov}[r_1,r_3]\)
\(w_1 w_4 \text{cov}[r_1,r_4]\)
\(w_1 w_5 \text{cov}[r_1,r_5]\)
\(w_2 w_1 \text{cov}[r_2,r_1]\)
\(w_2^2 \text{var}[r_2]\)
\(w_2 w_3 \text{cov}[r_2,r_3]\)
\(w_2 w_4 \text{cov}[r_2,r_4]\)
\(w_2 w_5 \text{cov}[r_2,r_5]\)
\(w_3 w_1 \text{cov}[r_3,r_1]\)
\(w_3 w_2 \text{cov}[r_3,r_2]\)
\(w_3^2 \text{var}[r_3]\)
\(w_3 w_4 \text{cov}[r_3,r_4]\)
\(w_3 w_5 \text{cov}[r_3,r_5]\)
\(w_4 w_1 \text{cov}[r_4,r_1]\)
\(w_4 w_2 \text{cov}[r_4,r_2]\)
\(w_4 w_3 \text{cov}[r_4,r_3]\)
\(w_4^2 \text{var}[r_4]\)
\(w_4 w_5 \text{cov}[r_4,r_5]\)
\(w_5 w_1 \text{cov}[r_5,r_1]\)
\(w_5 w_2 \text{cov}[r_5,r_2]\)
\(w_5 w_3 \text{cov}[r_5,r_3]\)
\(w_5 w_4 \text{cov}[r_5,r_4]\)
\(w_5^2 \text{var}[r_5]\)
Why does covariance dominate with large \(N\)?
-Consider an \(N\)-asset, equal-weighted portfolio \((w=1/N\))
-Assume all assets have the same variance \((\sigma^2_{\text{all}})\)
-Assume all pairs of assets have the same covariance \((\overline{\text{cov}})\)
-What is the variance of the portfolio?
-Diversification eliminates some, but not all, of the risk of individual assets.
-In large portfolios, \(\text{var}[r_i]\)’s effectively diversified away, but not \(\text{cov}[r_i,r_j]\)’s.
-Diversifiable, non-systematic, idiosyncratic risk vs. non-diversifiable, systematic, market risk
Diversification curves
INSERT PLOTS OR WEBPAGE OF EMPIRICAL DIVERSIFICATION CURVES?
Preferences
Which return series do you prefer?
Code
import numpy as npimport pandas as pdfrom scipy.stats import normimport plotly.express as pximport plotly.io as piopio.renderers.default='notebook'# Parametersmn1 =0.15sd1 =0.10mn2 =0.35sd2 =0.10T =50# Generate datarv1 = norm(loc=mn1, scale=sd1).rvs(T)rv2 = norm(loc=mn2, scale=sd2).rvs(T)time = np.arange(T)df = pd.DataFrame(data={'time': time, 'ret1': rv1, 'ret2': rv2})# Plot datafig = px.line(df,x='time', y=['ret1', 'ret2'])fig.update_layout(yaxis_title='Return', xaxis_title='', legend_title_text='',)fig.show()
Which return series do you prefer?
Code
import numpy as npimport pandas as pdfrom scipy.stats import normimport plotly.express as pximport plotly.io as piopio.renderers.default='notebook'# Parametersmn1 =0.20sd1 =0.10mn2 =0.20sd2 =0.40T =50# Generate datarv1 = norm(loc=mn1, scale=sd1).rvs(T)rv2 = norm(loc=mn2, scale=sd2).rvs(T)time = np.arange(T)df = pd.DataFrame(data={'time': time, 'ret1': rv1, 'ret2': rv2})# Plot datafig = px.line(df,x='time', y=['ret1', 'ret2'])fig.update_layout(yaxis_title='Return', xaxis_title='', legend_title_text='',)fig.show()
Which return series do you prefer?
Code
import numpy as npimport pandas as pdfrom scipy.stats import normimport plotly.express as pximport plotly.io as piopio.renderers.default='notebook'# Parametersmn1 =0.20sd1 =0.10mn2 =0.40sd2 =0.40T =50# Generate datarv1 = norm(loc=mn1, scale=sd1).rvs(T)rv2 = norm(loc=mn2, scale=sd2).rvs(T)time = np.arange(T)df = pd.DataFrame(data={'time': time, 'ret1': rv1, 'ret2': rv2})# Plot datafig = px.line(df,x='time', y=['ret1', 'ret2'])fig.update_layout(yaxis_title='Return', xaxis_title='', legend_title_text='',)fig.show()
-For a given target return, an efficient portfolio is the set of portfolio weights that minimize the portfolio’s risk (variance or standard deviation).
-Mathematically, choose portfolio weights to solve the following constrained optimization problem for each target return \(\mu_{\text{target}}\):
Alternatively, some investors have a either a target expected return or target standard deviation.
If we have a target expected return, solve for \(w_{\text{tangency}}\):
\[ w_i = \frac{\text{Value of Position}}{\text{Total Portfolio Value}} \]
Some investors cannot or choose not to short sell.
Sensitivity of portfolio weights to inputs (aka Garbage In=Garbage Out)
FROM YUHANG’S SLIDES, HOW FRENCH PORTFOLIO WGT CHANGES BASED ON THE INPUTS
COULD ALSO BE A HOMEWORK ASSIGNMENT
Inputs Must Be Estimated!1
Mean-variance optimization is quite sensitive to the input parameters.
Inputs:
-Set of expected returns for assets <br />
-Set of variances for assets <br />
-Set of covariances (correlations) across assets <br />
How good are we at estimating these things?
The Error-Maximization Problem
Mean-variance portfolio optimization:
-Will tilt too heavily toward assets with estimated expected returns above true expected returns \((\hat{\mu}>\mu)\)
-Will tilt too heavily toward assets with diversification benefits greater than true benefits \((\widehat{\text{cov}}_{ij}<\text{cov}_{ij})\)
-Often tries to short assets with diversification benefits lower than true benefits \((\widehat{\text{cov}}_{ij}>\text{cov}_{ij})\)
Performance of Historical Plug-in Estimators
ADD WEBPAGE DOING SIMULATION IN KEVIN’S NOTES BASED ON 5-ASSET ALLOCATION EXAMPLE
Strategies Dealing with Estimation Error
Shrinkage
Shrink extreme inputs toward some more moderate input
Short-selling constraints:
Prevent hedging positions due to overestimated covariances or underestimated expected returns
Maximum position limits
Prevents overweighting due to overestimated \(E[r]\) or underestimated covariances
Strategies Dealing with Estimation Error
Apply models to infer expected returns
Use market wgts to back out \(E[r_i]\)’s via CAPM (Black-Litterman)
Use more advanced factor models to estiamte \(E[r_i]\)’s
Don’t even try to estimate some inputs
Global minimum variance: assume all \(E[r_i]\)’s equal
Risk parity: assume all \(E[r_i]\)’s equal and all \(\rho_{ij}=0\)
Equal-weighted portfolio: assume all \(E[r_i]\)’s, \(\text{sd}[r_i]\)’s equal; all \(\rho_{ij}=0\)
Historical Performance
LINK TO OPTIMAL PORTFOLIOS OF STOCKS, BONDS, AND GOLD?